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MATHEMATICS Lecture. 4
Chapter .8
TECHNIQUES OF INTEGRATION
By
Dr. Mohammed Ramidh
Dr. Mohammed Ramidh
TECHNIQUES OF INTEGRATION
OVERVIEW The Fundamental Theorem connects antiderivatives and the definite integral. Evaluating the indefinite integral, In this chapter we study a number of important techniques for finding indefinite integrals of more complicated functions than those seen before.
8.1 Basic Integration Formulas.
In this section we present several algebraic or substitution methods to help us use this table 8-1.
Dr. Mohammed Ramidh
Dr. Mohammed Ramidh
Dr. Mohammed Ramidh
Dr. Mohammed Ramidh
EXAMPLE 1: Evaluate
Dr. Mohammed Ramidh
Dr. Mohammed Ramidh
EXAMPLE 2: Evaluate
EXAMPLE 3: Expanding a Power and Using a Trigonometric Identity
Evaluate.
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EXAMPLE 4: Evaluate,
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EXAMPLE 5: Separating a Fraction, Evaluate
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EXAMPLE 6: Integral of —Multiplying by a Form of 1, Evaluate.
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EXERCISES 8.1 1. Evaluate each integral in Exercises 1–18 by using a substitution to
reduce it to standard form.
2. Evaluate each integral in Exercises 37–42 by completing the square
and using a substitution to reduce it to standard form.
3. Evaluate each integral in Exercises 43–46 by using trigonometric
identities and substitutions to reduce it to standard form.
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4. Evaluate each integral in Exercises 53–55 by separating the fraction
and using a substitution (if necessary) to reduce it to standard form.
5. Evaluate each integral in Exercises 57–62 by multiplying by a form of 1 and using a substitution (if necessary) to reduce it to standard form.
6. Evaluate each integral in Exercises 63–70 by eliminating the square
root.
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8.2 Integration by Parts In this section, we describe integration by parts and show how to apply it.
Product Rule in Integral Form
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EXAMPLE 1: Using Integration by Parts, Find.
Dr. Mohammed Ramidh
Dr. Mohammed Ramidh
EXAMPLE 2: Find.
EXAMPLE 5: Evaluate.
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˝Integration by Parts Formula for Definite Integrals
EXAMPLE 4:
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Dr. Mohammed Ramidh
2.
3.
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4.
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5.
Dr. Mohammed Ramidh
Tabular Integration tabular integration is illustrated in the following examples.
EXAMPLE 5 : Using Tabular Integration , Evaluate
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EXAMPLE 6: Using Tabular Integration, Evaluate
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EXERCISES 8.2 1. Integration by Parts, Evaluate the integrals in Exercises 1–22.
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8.3 Integration of Rational Functions by Partial Fractions
This section shows how to express a rational function (a quotient of polynomials) as a sum of simpler fractions, called partial fractions, which are easily integrated. For example, the rational function (5x - 3) ⁄ (x2 - 2x – 3)can be rewritten as
The method for rewriting rational functions as a sum of simpler fractions is called the method of partial fractions. In the case of the above example, it consists of finding constants A and B such that
To find A and B, we first clear Equation (1) of fractions, obtaining
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To integrate the rational function,
˝General Description of the Method:
Success in writing a rational function ƒ(x) ⁄ g(x) as a sum of partial fractions depends on two things:
• The degree of ƒ(x) must be less than the degree of g(x).
• We must know the factors of g(x).
Here is how we find the partial fractions of a proper fraction ƒ(x)⁄ g(x) when the factors of g are known.
Dr. Mohammed Ramidh
Dr. Mohammed Ramidh
EXAMPLE 1: Evaluate , using partial fractions.
Solution : The partial fraction decomposition has the form
To find the values of the undetermined coefficients A, B, and C we clear fractions and get
So we equate coefficients of like powers of x obtaining
Dr. Mohammed Ramidh
Dr. Mohammed Ramidh
Dr. Mohammed Ramidh
EXAMPLE 2: Evaluate,
Solution: First we express the integrand as a sum of partial fractions with undetermined coefficients.
Equating coefficients of corresponding powers of x gives
EXAMPLE 3: Integrating an Improper Fraction, Evaluate
Solution : First we divide the denominator into the numerator to get
a polynomial plus a proper fraction.
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Then we write the improper fraction as a polynomial plus a proper fraction.
We found the partial fraction decomposition of the fraction on the right in the opening example, so
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EXAMPLE 4: Integrating with an Irreducible Quadratic Factor in the Denominator, Evaluate using partial fractions.
Solution: The denominator has an irreducible quadratic factor as well as a repeated linear factor, so we write
Clearing the equation of fractions gives
Equating coefficients of like terms gives
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We solve these equations simultaneously to find the values of A, B, C, and D:
We substitute these values into Equation (2), obtaining
Dr. Mohammed Ramidh
Dr. Mohammed Ramidh
1.
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2.
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3.
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4.
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5.
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6.
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EXERCISES 8.3
1. Expand the quotients in Exercises 1–8 by partial fractions.
2. In Exercises 9–14, express the integrands as a sum of partial fractions and evaluate the integrals.
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8.4 Trigonometric Integrals
Trigonometric integrals involve algebraic combinations of the six basic trigonometric functions.
1. Products of Powers of Sines and Cosines
We begin with integrals of the form:
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EXAMPLE 1: m is Odd, Evaluate
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Dr. Mohammed Ramidh
Dr. Mohammed Ramidh
Dr. Mohammed Ramidh
Integrals of Powers of tan x and sec x
2. Products of Sines and Cosines
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Dr. Mohammed Ramidh
Dr. Mohammed Ramidh
Dr. Mohammed Ramidh
Dr. Mohammed Ramidh
Dr. Mohammed Ramidh
Dr. Mohammed Ramidh
EXERCISES 8.4 1. Products of Powers of Sines and Cosines, Evaluate the integrals
in Exercises 1–10.
.
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2. Products of Sines and Cosines ,Evaluate the integrals in Exercises 33–38
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